Double-Angle Formula & Half-Angle Formula

In this lesson, we learn how to use the double angle formulas and the half-angle formulas to solve trigonometric equations and to prove trigonometric identities. We will also prove the double angle formulas and the half-angle formulas.

• Using Double Angle Identities To Solve Equations Click here
• Prove Trigonometry Identities Using Double Angles Click here
• Proof Of The Double Angle Formulae Click here
• Using Half-Angle Formulas Click here
• Proof of the Half-Angle Formulas Click here

The following figures give the Double-Angle Formulas and Half-Angle Formulas. Scroll down the page for more examples and solutions on how to use, derive and proof the Double-Angle Formulas and Half-Angle Formulas.

Using Double Angle Identities To Solve Equations

Using Double Angle Identities to Solve Equations, Example 1.
This video uses some double angle identities for sine and/or cosine to solve some equations.

Example:
cos(4x) − 3cos(2x) = 4

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Using Double Angle Identities to Solve Equations, Example 2

Example:
sin(2t) = sin(t)

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Using Double Angle Identities to Solve Equations, Example 3

Example:
sin(2t) + 4sin(t) + 2cos(t) = −4

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Example:
Given , find
a) sin 2θ
b)

Solution:
a) sin 2θ = 2 sin θ cos θ

Prove Trigonometry Identities Using Double Angles

Trigonometry Identities - Double Angles (1)

Example:
(1 − cos 2x)/sin 2x = tan x

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Trigonometry Identities - Double Angle (2)

Example:
Prove tan x + cos x = 2 cosec 2x

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Double-Angle Formulas

A few examples that use double-angle formulas from trigonometry.

Example:

1. If cos x = 1/√10 with x in quadrant IV, find sin 2x
2. Graph y = 4 - 8 sin²x
3. Verify sin60° = 2sin30°cos30°

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Proof Of The Double Angle Formulae

Trigonometry - Proof of the Double Angle Formulae from the Addition Formulae.

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How to derive the double angle identities for sine and cosine?

Example:
If cosθ = -4/5 and sinθ > 0, find
a) sin2θ
b) cos2θ

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Using Half-Angle Formulas

Half-Angle Formulas
A few problems involving half-angle formulas in trigonometry.

Example:
Let sinA = 4/5 with A in Quadrant II. Find

1. sin A/2
2. cos 2A
3. sec 2A

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Half Angle Identities to Evaluate Trigonometric Expressions, Example 1

This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.

Example:
Find the exact value of sin(22.5°)

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Half Angle Identities to Evaluate Trigonometric Expressions, Example 2

This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.

Example:
Find the exact value of tan(105°)

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Half Angle Identities to Evaluate Trigonometric Expressions, Example 3

This video gives some half angle identities and shows how they can be used to solve some trigonometric equations.

Find sin(a/2) if cos a = 3/5 for 0° ≤ a ≤ 90°

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Proof of the Half-Angle Formulas

This video derives the half angle trigonometry identities for cosine, sine and tangent.

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